To determine which of the given sets of numbers is NOT a Pythagorean triple, we need to check if each set satisfies the Pythagorean theorem. The Pythagorean theorem states that for a set of three positive integers \(a, b, c\) (where \(a \leq b \leq c\)), the set is a Pythagorean triple if and only if:
[tex]\[a^2 + b^2 = c^2\][/tex]
Let's examine each set step-by-step:
1. Set A: \(2, 3, 4\)
- Let's assign \(a = 2\), \(b = 3\), and \(c = 4\).
- Calculate \(a^2 + b^2: 2^2 + 3^2 = 4 + 9 = 13\).
- Calculate \(c^2: 4^2 = 16\).
- Check if \(a^2 + b^2 = c^2: 13 \neq 16\).
- Conclusion: Set A (\(2, 3, 4\)) is NOT a Pythagorean triple.
2. Set B: \(3, 4, 5\)
- Let's assign \(a = 3\), \(b = 4\), and \(c = 5\).
- Calculate \(a^2 + b^2: 3^2 + 4^2 = 9 + 16 = 25\).
- Calculate \(c^2: 5^2 = 25\).
- Check if \(a^2 + b^2 = c^2: 25 = 25\).
- Conclusion: Set B (\(3, 4, 5\)) is a Pythagorean triple.
3. Set C: \(6, 8, 10\)
- Let's assign \(a = 6\), \(b = 8\), and \(c = 10\).
- Calculate \(a^2 + b^2: 6^2 + 8^2 = 36 + 64 = 100\).
- Calculate \(c^2: 10^2 = 100\).
- Check if \(a^2 + b^2 = c^2: 100 = 100\).
- Conclusion: Set C (\(6, 8, 10\)) is a Pythagorean triple.
4. Set D: \(5, 12, 13\)
- Let's assign \(a = 5\), \(b = 12\), and \(c = 13\).
- Calculate \(a^2 + b^2: 5^2 + 12^2 = 25 + 144 = 169\).
- Calculate \(c^2: 13^2 = 169\).
- Check if \(a^2 + b^2 = c^2: 169 = 169\).
- Conclusion: Set D (\(5, 12, 13\)) is a Pythagorean triple.
After evaluating each set, we find that the set of numbers \(2, 3, 4\) does not satisfy the Pythagorean theorem.
Therefore, the answer is:
A. [tex]\(2, 3, 4\)[/tex]
